Quantity Basics

• Quantity (Q): a measurable amount of something. Requires a unit (u) and a number (n) to completely communicate a quantity (3 pencils; 5.2 g; 25 m). Q = n*u

• Unit (u): the size of a quantity we call 1 for a given measurement or comparison. The scientific community has sets of standardized units that we have defined, such as 1 kg or 1 m. Unit (un-it) comes from the Latin for 1. It is one "it"! We can also consider any quantity to be a unit for the purpose of comparing sizes of two quantities. (See Number, below).

• Number (n): a ratio between a quantity and its unit used to compare their sizes (magnitudes).

  • • Ex: 3 pencils compared to 1 pencil expresses the number 3 (no unit). A number communicates relative size of either two physical quantities (science) or of two abstract quantities (two geometric line lengths or two angles between intersecting lines - pure math). n = Q/u

Relationships Between Quantities

• Connected (Related) Quantities: two measurements made on the same substance, event, or system that MUST change together in some way due to the nature of that substance or system. We will consider two primary ways two quantities can be related (connected) to each other:

  • Ratio Relationship: expresses a connection between two quantities in nature such that when one doubles, the other also doubles; the ratio between them is constant.

    • • Ex: An object moving at constant speed has a necessary connection between distance and time traveled (5m/1s) because they both describe the same object in our space-time universe. If time doubles to 2s, then distance also must double to 10m!

  • Product Relationship: a quantitative connection between two quantities in nature such that when one doubles, the other cuts in half; the product between them is constant.

    • • Ex: Two rectangles can have the same area, but have different side lengths. Length are width are connected by a product relationship (A = l x w). A rectangle with l = 5m and w = 4m has the same area (20 m^2) as one where l = 10 m and w = 2 m. Both have an area of 20 m^2. To keep the same area, if length is increased, width must go down by the same proportion (2*L --> 1/2*W; 3*L --> 1/3*L; etc.). A product relationship can also be called an inverse relationship.

Math Symbols in Science

• Equal sign (=): communicates that two quantitative statements are either

  • • identical: same in amount and structure, or the same by definition.

      • • Ex: (2g + 3g) = (2g + 3g); or, n = Q/u.

    • equivalent: same in total amount, but not organized/structured the same way.

      • • Ex: a rectangle that is 3m*4m forms an equivalent (but not identical) area to one that is 6m*2m

        • Ex: F =(m/a): force has the same magnitude & direction as the mass/acceleration ratio, but F and m/a are connected to each other, but not identical things.

• Division sign (/): indicates we are seeing one of two connections between two quantities:

  • • Size comparison between two quantities of the same type and unit

      • • Ex: 10g/2g = 5 (or, Q/u = n), which means 10g is 5 times bigger than 2g

    • Ratio relationship (see definition above) between two quantities inherently connected by definition or by nature.

      • • Ex: 100cm/1m; these are equivalent expressions of the same length because of the definitions of the units.

        • Ex: 5m/1s; distance is connected to time for this moving object - every 1 sec the object moves 5 meters; the object always has a specific location in both space and time. The 4-dimensional space-time nature of our universe ALWAYS causes position and time to be inherently connected for a given object in the universe.

• Multiplication sign (* or x): indicates three different scenarios in science:

  • • two quantities are connected via a product relationship (ex: L*W; see above);

    • the magnitude of a quantity is being changed by a number: Q1 * n = Q2

      • • Ex: 3g * 2 = 6g

    • a ratio relationship is acting on one quantity to find the value of the other quantity: Q1 * R = Q2

      • • Ex: 5m * 100cm/1m = 500cm

        • Ex: 10s * 5m/s = 50m

• Addition sign (+): used to join two or more quantities of the same kind, like adding the side lengths of a rectangle to find its perimeter (P = S1 + S2 + S3 + S4). Quantities that are not of the same kind cannot be joined.

  • • Ex: 3 pencils + 2 pencils makes a single group of 5 pencils, but 3 pencils + 2 erasers does not equal 5 of anything; it is still a group of 3 pencils and 2 erasers. If you are simply counting objects without regard for what they are, it would be written 2 objects + 3 objects = 5 objects.

• Subtraction sign (-): used to express a change in a quantity in a system.

  • • Ex: If a system had a mass of 10g and then was found to be 6g, then we would write: 6g - 10g = -4g. The system changed by (lost) 4g of mass.

• Positive/Negative sign (+, - attached to a single quantity or relationship):

  • • Quantities: In the example above, we showed a system that lost mass. The calculated change had a negative sign (-) on it to indicate the direction of change in the system (mass was lost). Putting + or - on a single quantity only makes sense if we are expressing the magnitude and direction of a change (a difference). Ex: the mass changed by -4 g (ie, lost 4 g).

    • Relationships: The + and - sign may also be used on a ratio relationship to indicate whether a positive change in one produces a positive or negative change in the other. If a velocity is -5m/1s, then a time difference of +1s will be associated with a displacement of -5m (or, the object moves 5m in the negative direction on the position axis.)